Digital Object Identifier (DOI) 10.1007/s00205-005-0402-5 Arch. Rational Mech. Anal. 180 (2006) 331–398 The Singular Set of Minima of Integral Functionals Jan Kristensen & Giuseppe Mingione Communicated by V. Šverák Abstract In this paper we provide upper bounds for the Hausdorff dimension of the singular set of minima of general variational integrals   F(x,v,Dv) dx, where F is suitably convex with respect to Dv and Hölder continuous with respect to (x,v). In particular, we prove that the Hausdorff dimension of the singular set is always strictly less than n, where  ⊂ R n . 1. Introduction and Results We consider integral functionals of the type F [ v,A] :=  A F(x,v,Dv)dx defined for Sobolev maps v ∈ W 1,p loc (, R N ) and open sets A whose closure is com- pact and contained in . Here for n, N ≧ 2,  is a bounded open set in R n , p ≧ 2 and F :  × R N × R nN → R is a suitably regular integrand. A local minimizer of the functional F is a map u ∈ W 1,p loc (, R N ) such that F [ u, A] ≦ F [ v,A] , whenever A ⊂⊂  and u - v ∈ W 1,p 0 (A, R N ). A classical problem in the Cal- culus of Variations consists of studying the regularity properties of such maps.